Baryon Structure from Lattice QCD with 2+1 Flavours of Wilson Quarks

**Principal Investigator:**

Harvey Meyer

**Affiliation:**

Johannes Gutenberg University Mainz

**Local Project ID:**

chmz36

**HPC Platform used:**

JUQUEEN and JUWELS of JSC

**Date published:**

The Higgs boson, discovered at the Large Hadron Collider (LHC) in 2012, was the last missing building block of the Standard Model (SM) of particle physics. This theory simultaneously describes the electromagnetic, the weak and the strong force at a fundamental level. However, the SM cannot possibly be the ultimate description of nature, since it not only fails to account for gravity, but also lacks any viable particle candidate for dark matter, whose existence is inferred from astronomical observations. Therefore the search for particles and phenomena beyond the SM continues to be actively pursued.

At the same time, making predictions based on the SM often represents a challenging computational task. This is particularly so in the case of the strong force, where the still mysterious property of quark confinement (“one quark is never seen alone”) makes quantitative predictions extremely difficult.

The only first-principles formulation of QCD that allows for systematically reducing the errors is lattice QCD, which has had a significant scientific impact by providing precise predictions for the masses and properties of hadrons [1], the bound states of quarks. Protons, neutrons and pions are examples of hadrons, but many more types of hadrons exist.

Nevertheless, many challenges remain for lattice QCD, as there are a number of quantities that so far could not be studied with the desired level of control of statistical and systematic errors. This is especially true for the structural properties of baryons such as protons and neutrons. Baryons are those hadrons which obey the Pauli exclusion principle.

One prominent observable under scrutiny in this project is the “scalar matrix element” of the proton. The scalar matrix element provides a quantitative answer to the question of “How much would the proton mass change if the quark masses changed by a small amount?”. More practically, it plays a central role in interpreting the results of dark-matter direct-detection experiments [2] if one assumes that the dark-matter particle interacts with atomic nuclei via the exchange of a Higgs boson. A tension of roughly three standard deviations [3] has emerged between lattice QCD results and phenomenological determinations of the light-quark scalar matrix element. This tension demands clarification.

In 2019, we published a set of results for similar proton matrix elements [4] as part of project chmz36. In particular, we calculated its axial charge with a total uncertainty of 3% and found a value compatible with its experimental determination, an important validation step. Further proton-structure observables currently investigated in this project are the proton charge and magnetic radii. If one thinks of the proton as a cloud containing electric charges and currents, these radii answer the question of how extended the cloud is. Similarly, the axial radius is related to the quark-spin distribution in the proton, and is currently a limiting factor in predicting the scattering probability of neutrinos on atomic nuclei [5], which is needed to interpret the measurements of upcoming long-baseline neutrino oscillation experiments (DUNE, T2HyperK).

All these important scientific goals can only be realized by performing lattice QCD simulations on fine lattices at realistic values of the quark masses. In lattice QCD, space and time are approximated by a cubic lattice. Simulations are performed at several lattice spacings before the physical observables are extrapolated to the continuum. Since the volume of the lattice must be sufficiently large to contain the proton without “squeezing’’ it, a small lattice spacing implies a very large lattice and a high computational cost. Performing calculations at quark masses as light as in nature also implies a high computational cost. The lattice used in this project has a size of 96x96x96x192 and the quark masses are set to realistic values. The calculations are performed on the JUWELS cluster module using 12288 cores in parallel with MPI instructions.

Figure 1 shows an example of the preliminary results achieved. The quantity displayed is the scalar form factor as a function of the momentum transfer Q^{2}. A remark for the experts: only the quark-connected contribution is included here. The value at Q^{2}=0 is the scalar matrix element described above. The plot shows four sets of data points, corresponding to the form factor being extracted from QCD correlation functions at different separations t_{sep}. An important source of systematic error is suppressed at large t_{sep}, however, as visible in the figure, the statistical error then increases. Presently, the project is ongoing in order to further reduce the uncertainty of the calculation. The project is expected to be completed in 2021.

**Current project contributors**

Andria Agadjanov, Dalibor Djukanovic, Georg von Hippel, Konstantin Ottnad, Tobias Schulz, Hartmut Wittig

**References**

[1] S.Aoki et al. [Flavour Lattice Averaging Group], “FLAG Review 2019: Flavour Lattice Averaging Group (FLAG),'' Eur. Phys. J. C 80 (2020) no.2, 113 doi:10.1140/epjc/s10052-019-7354-7 [arXiv:1902.08191 [hep-lat]].

[2] J.M. Cline, K. Kainulainen, P. Scott and C. Weniger, “Update on scalar singlet dark matter,'' Phys. Rev. D 88 (2013), 055025 doi:10.1103/PhysRevD.88.055025 [arXiv:1306.4710 [hep-ph]].

[3] M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U.G. Meißner, "Remarks on the pion–nucleon σ-term,'' Phys. Lett. B 760 (2016), 74-78 doi:10.1016/j.physletb.2016.06.038 [arXiv:1602.07688 [hep-lat]].

[4] T. Harris, G. von Hippel, P. Junnarkar, H.B.Meyer, K. Ottnad, J. Wilhelm, H. Wittig and L. Wrang, “Nucleon isovector charges and twist-2 matrix elements with $N_f=2+1$ dynamical Wilson quarks,'' Phys. Rev. D 100 (2019) no.3, 034513 doi:10.1103/PhysRevD.100.034513 [arXiv:1905.01291 [hep-lat]].

[5] A.S. Kronfeld et al. [USQCD], “Lattice QCD and Neutrino-Nucleus Scattering,'' Eur. Phys. J. A 55 (2019) no.11, 196 doi:10.1140/epja/i2019-12916-x [arXiv:1904.09931 [hep-lat]].

**Scientific Contact**

Univ.-Prof. Dr. Harvey B. Meyer

Institute for Nuclear Physics

Johannes Gutenberg University Mainz

Johann-Joachim-Becher Weg 45, D-55099 Mainz (Germany)

e-mail: meyerh [@] uni-mainz.de

*Local project ID: chmz36*

*September 2020*